Social Welfare in One-Sided Matchings: Random Priority and Beyond

Abstract

Abstract. We study the problem of approximate social welfare maximization (without money) in onesided matching problems when agents have unrestricted cardinal preferences over a finite set of items. Random priority is a very well-known truthful-in-expectation mechanism for the problem. We prove that the approximation ratio of random priority is Θ(n −1/2 ) while no truthful-in-expectation mechanism can achieve an approximation ratio better than O(n −1/2 ), where n is the number of agents and items. Furthermore, we prove that the approximation ratio of all ordinal (not necessarily truthful-in-expectation) mechanisms is upper bounded by O(n −1/2 ), indicating that random priority is asymptotically the best truthful-in-expectation mechanism and the best ordinal mechanism for the problem.

Truthful Mechanisms for Matching and Clustering in an Ordinal World

“…A majority of the papers in this domain deal with mechanisms that elicit agent utilities, specifically for one-sided matchings, assignments and facility location problems that are somewhat different from the graph problems we are interested in. The notable exceptions are the recent papers on truthful, ordinal mechanisms for one-sided matchings [16,8] and general allocation problems [2]. While [16] looks at normalized agent utilities and shows that no ordinal algorithm can provide an approximation factor better than Θ( √ N ), [8] considers minimum cost metric matching under a resource augmentation framework.…”

“…The notable exceptions are the recent papers on truthful, ordinal mechanisms for one-sided matchings [16,8] and general allocation problems [2]. While [16] looks at normalized agent utilities and shows that no ordinal algorithm can provide an approximation factor better than Θ( √ N ), [8] considers minimum cost metric matching under a resource augmentation framework. The main differences between our work and these two papers are (1) we consider two-sided matching instead of one-sided, as well as other clustering problems, as well as non-truthful algorithms with better approximation factors, and (2) we consider maximization objectives in which users attempt to maximize their utility instead of minimize their cost.…”

Average-case Analysis of the Assignment Problem with Independent Preferences

“…Here we plugged in the fact that SW OPT (A) ≤ n, for any instance A. In addition, as shown in [20], r worst ∼ Θ( √ n), there exists a constant c 1 , such that r worst ≤ c 1 · √ n, for sufficiently large n. Also, it is obvious that Pr{SW RP (A) > λ} ≤ 1. Now, we plug in these fact and the results established in Lemma 1.…”

“…where n is the number of agents and items, and it is asymptotically the best amongst all truthful mechanisms [20]. This negative result was considered a cautionary tale discouraging the wide applications of Random Priority.…”

Subset Selection Via Implicit Utilitarian Voting

“…Approximating utilitarian social welfare given ordinal information has also been studied in mechanism design. Filos-Ratsikas, Frederiksen, and Zhang (2014) apply this notion for finding matchings in graphs; Krysta, Manlove, Rastegari, and Zhang (2014) apply it to the house allocation problem; and Chakrabarty and Swamy (2014) study approximation of welfare under the assumption that an agent's utility for an alternative depends only on the rank of the alternative in the agent's preference order, i.e., when the utilities of all agents are dictated by a common underlying positional scoring rule. We refer the reader to the paper by Boutilier et al (2015, Section 1.2) for a thorough discussion of work (in philosophy, economics, and social choice theory) related to implicit utilitarian voting more broadly.…”